10 154

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-4 t+7-4 t^{-1} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+6 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 13, 4 }
Jones polynomial	[math]\displaystyle{ q^{12}-2 q^{11}+2 q^{10}-3 q^9+2 q^8-2 q^7+2 q^6+q^3 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-6} +6 z^4 a^{-6} +9 z^2 a^{-6} -2 z^2 a^{-8} -2 z^2 a^{-10} +4 a^{-6} -2 a^{-8} -2 a^{-10} + a^{-12} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-10} +z^8 a^{-12} +z^7 a^{-9} +3 z^7 a^{-11} +2 z^7 a^{-13} +z^6 a^{-6} -5 z^6 a^{-10} -3 z^6 a^{-12} +z^6 a^{-14} -6 z^5 a^{-9} -15 z^5 a^{-11} -9 z^5 a^{-13} -6 z^4 a^{-6} -2 z^4 a^{-8} +7 z^4 a^{-10} -z^4 a^{-12} -4 z^4 a^{-14} -2 z^3 a^{-7} +9 z^3 a^{-9} +21 z^3 a^{-11} +10 z^3 a^{-13} +9 z^2 a^{-6} +5 z^2 a^{-8} -5 z^2 a^{-10} +2 z^2 a^{-12} +3 z^2 a^{-14} +3 z a^{-7} -3 z a^{-9} -10 z a^{-11} -4 z a^{-13} -4 a^{-6} -2 a^{-8} +2 a^{-10} + a^{-12} }[/math]
The A2 invariant	[math]\displaystyle{ q^{-10} + q^{-12} + q^{-14} +2 q^{-16} +2 q^{-18} + q^{-22} - q^{-24} - q^{-26} -2 q^{-28} -2 q^{-30} - q^{-34} + q^{-36} + q^{-38} }[/math]
The G2 invariant	[math]\displaystyle{ q^{-50} + q^{-52} +2 q^{-56} + q^{-58} +3 q^{-62} + q^{-66} +3 q^{-68} -2 q^{-70} +3 q^{-72} +3 q^{-74} -4 q^{-76} +7 q^{-78} -5 q^{-80} +2 q^{-82} +5 q^{-84} -7 q^{-86} +9 q^{-88} -4 q^{-90} -4 q^{-92} +8 q^{-94} -5 q^{-96} +7 q^{-100} -10 q^{-102} +8 q^{-104} -2 q^{-106} -6 q^{-108} +8 q^{-110} -9 q^{-112} +6 q^{-114} -5 q^{-116} -2 q^{-118} + q^{-120} -3 q^{-122} -6 q^{-126} -2 q^{-130} -2 q^{-132} + q^{-134} -4 q^{-136} -2 q^{-138} +8 q^{-140} -10 q^{-142} +6 q^{-144} -7 q^{-148} +14 q^{-150} -10 q^{-152} +6 q^{-154} +4 q^{-156} -5 q^{-158} +8 q^{-160} -3 q^{-162} - q^{-164} +6 q^{-166} -4 q^{-168} +3 q^{-172} -5 q^{-174} +6 q^{-176} -4 q^{-178} - q^{-180} + q^{-182} -3 q^{-184} +2 q^{-186} - q^{-188} + q^{-190} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_154.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{-5} + q^{-7} +2 q^{-11} - q^{-17} - q^{-19} - q^{-23} + q^{-25} }[/math]
2	[math]\displaystyle{ q^{-10} + q^{-12} + q^{-14} +3 q^{-20} +2 q^{-22} -2 q^{-24} + q^{-26} +2 q^{-28} -2 q^{-30} -3 q^{-32} -2 q^{-38} -2 q^{-40} +2 q^{-42} - q^{-44} -2 q^{-46} +4 q^{-48} -2 q^{-52} +2 q^{-54} +2 q^{-56} - q^{-58} -2 q^{-60} + q^{-62} +2 q^{-64} -2 q^{-66} - q^{-68} + q^{-70} }[/math]
3	[math]\displaystyle{ q^{-15} + q^{-17} + q^{-19} + q^{-21} +3 q^{-29} +3 q^{-31} +3 q^{-33} -2 q^{-35} -4 q^{-37} - q^{-39} +6 q^{-41} +8 q^{-43} -6 q^{-45} -15 q^{-47} -2 q^{-49} +14 q^{-51} +6 q^{-53} -16 q^{-55} -12 q^{-57} +8 q^{-59} +10 q^{-61} -2 q^{-63} -8 q^{-65} +4 q^{-69} +3 q^{-71} -2 q^{-73} -3 q^{-75} + q^{-77} +6 q^{-79} -6 q^{-83} +2 q^{-85} +11 q^{-87} -12 q^{-91} -4 q^{-93} +12 q^{-95} +8 q^{-97} -11 q^{-99} -12 q^{-101} +2 q^{-103} +11 q^{-105} +4 q^{-107} -6 q^{-109} -6 q^{-111} +5 q^{-115} +4 q^{-117} - q^{-119} -4 q^{-121} -3 q^{-123} +3 q^{-125} +3 q^{-127} -2 q^{-131} - q^{-133} + q^{-135} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{-10} + q^{-12} + q^{-14} +2 q^{-16} +2 q^{-18} + q^{-22} - q^{-24} - q^{-26} -2 q^{-28} -2 q^{-30} - q^{-34} + q^{-36} + q^{-38} }[/math]
1,1	[math]\displaystyle{ q^{-20} +2 q^{-22} +2 q^{-24} +6 q^{-26} +5 q^{-28} +4 q^{-30} +8 q^{-32} -4 q^{-34} +12 q^{-36} -8 q^{-38} +2 q^{-40} -6 q^{-42} -14 q^{-44} +8 q^{-46} -20 q^{-48} +6 q^{-50} -8 q^{-52} -2 q^{-54} +8 q^{-56} -8 q^{-58} +14 q^{-60} -8 q^{-62} +12 q^{-64} -6 q^{-66} +3 q^{-68} -8 q^{-72} +16 q^{-74} -21 q^{-76} +22 q^{-78} -16 q^{-80} +12 q^{-82} -2 q^{-84} -8 q^{-86} +10 q^{-88} -12 q^{-90} +11 q^{-92} -8 q^{-94} +4 q^{-96} -2 q^{-98} + q^{-100} }[/math]
2,0	[math]\displaystyle{ q^{-20} + q^{-22} +2 q^{-24} + q^{-26} + q^{-28} +3 q^{-30} +5 q^{-32} +2 q^{-34} +2 q^{-36} +3 q^{-38} +3 q^{-40} + q^{-42} -2 q^{-44} -3 q^{-46} -3 q^{-48} -4 q^{-50} -4 q^{-52} -8 q^{-54} -6 q^{-56} -3 q^{-58} - q^{-60} - q^{-62} +3 q^{-64} +6 q^{-66} +4 q^{-68} +4 q^{-70} +2 q^{-74} + q^{-78} - q^{-80} - q^{-82} + q^{-84} -2 q^{-88} -3 q^{-90} + q^{-94} + q^{-96} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{-20} + q^{-22} +2 q^{-24} +3 q^{-26} +5 q^{-28} +3 q^{-30} +5 q^{-32} +2 q^{-34} -2 q^{-40} -3 q^{-42} -4 q^{-44} -4 q^{-46} -4 q^{-48} -6 q^{-50} -4 q^{-52} - q^{-54} +3 q^{-58} +6 q^{-60} +4 q^{-62} +2 q^{-64} +2 q^{-66} -2 q^{-68} - q^{-70} -2 q^{-72} - q^{-78} + q^{-80} }[/math]
1,0,0	[math]\displaystyle{ q^{-15} + q^{-17} + q^{-19} +3 q^{-21} +2 q^{-23} +2 q^{-25} + q^{-27} + q^{-29} - q^{-31} - q^{-33} -2 q^{-35} -2 q^{-37} -2 q^{-39} -2 q^{-41} - q^{-45} + q^{-47} + q^{-49} + q^{-51} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{-30} + q^{-32} +2 q^{-34} +4 q^{-36} +6 q^{-38} +6 q^{-40} +9 q^{-42} +7 q^{-44} +6 q^{-46} +5 q^{-48} +3 q^{-50} -3 q^{-52} -3 q^{-54} -3 q^{-56} -5 q^{-58} -9 q^{-60} -9 q^{-62} -8 q^{-64} -14 q^{-66} -13 q^{-68} -7 q^{-70} -4 q^{-72} - q^{-74} +9 q^{-76} +12 q^{-78} +11 q^{-80} +11 q^{-82} +10 q^{-84} +3 q^{-86} -3 q^{-88} -3 q^{-90} -3 q^{-92} -6 q^{-94} -4 q^{-96} + q^{-104} + q^{-106} }[/math]
1,0,0,0	[math]\displaystyle{ q^{-20} + q^{-22} + q^{-24} +3 q^{-26} +3 q^{-28} +2 q^{-30} +3 q^{-32} + q^{-34} + q^{-36} - q^{-38} - q^{-40} -2 q^{-42} -2 q^{-44} -2 q^{-46} -2 q^{-48} -2 q^{-50} -2 q^{-52} - q^{-56} + q^{-58} + q^{-60} + q^{-62} + q^{-64} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{-20} + q^{-22} +3 q^{-26} + q^{-28} + q^{-30} + q^{-32} +2 q^{-36} -2 q^{-38} +2 q^{-40} -3 q^{-42} +2 q^{-44} -2 q^{-46} -2 q^{-52} +3 q^{-54} -4 q^{-56} +3 q^{-58} -4 q^{-60} +2 q^{-62} -2 q^{-64} - q^{-70} +2 q^{-72} -2 q^{-74} +2 q^{-76} - q^{-78} + q^{-80} }[/math]
1,0	[math]\displaystyle{ q^{-30} + q^{-34} + q^{-36} + q^{-38} +3 q^{-42} +2 q^{-44} +2 q^{-46} + q^{-48} +2 q^{-50} +3 q^{-52} + q^{-54} -2 q^{-56} + q^{-58} +2 q^{-60} -3 q^{-64} -2 q^{-66} - q^{-70} -4 q^{-72} -4 q^{-74} - q^{-76} - q^{-78} -2 q^{-80} -4 q^{-82} -2 q^{-84} + q^{-86} + q^{-88} - q^{-90} +2 q^{-94} +4 q^{-96} +2 q^{-98} + q^{-102} +4 q^{-104} + q^{-106} -2 q^{-108} -2 q^{-110} + q^{-112} + q^{-114} -2 q^{-116} -2 q^{-118} + q^{-120} + q^{-122} - q^{-124} - q^{-126} + q^{-130} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{-30} + q^{-32} + q^{-34} +4 q^{-36} +3 q^{-38} +5 q^{-40} +5 q^{-42} +5 q^{-44} +3 q^{-46} +2 q^{-48} +2 q^{-50} -3 q^{-52} -4 q^{-56} -5 q^{-60} - q^{-62} -4 q^{-64} -3 q^{-66} -5 q^{-68} -6 q^{-70} -4 q^{-72} -5 q^{-74} + q^{-76} -3 q^{-78} +5 q^{-80} +2 q^{-82} +8 q^{-84} +2 q^{-86} +5 q^{-88} + q^{-92} -2 q^{-96} - q^{-98} -3 q^{-100} +2 q^{-102} -2 q^{-104} + q^{-106} - q^{-108} + q^{-110} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{-50} + q^{-52} +2 q^{-56} + q^{-58} +3 q^{-62} + q^{-66} +3 q^{-68} -2 q^{-70} +3 q^{-72} +3 q^{-74} -4 q^{-76} +7 q^{-78} -5 q^{-80} +2 q^{-82} +5 q^{-84} -7 q^{-86} +9 q^{-88} -4 q^{-90} -4 q^{-92} +8 q^{-94} -5 q^{-96} +7 q^{-100} -10 q^{-102} +8 q^{-104} -2 q^{-106} -6 q^{-108} +8 q^{-110} -9 q^{-112} +6 q^{-114} -5 q^{-116} -2 q^{-118} + q^{-120} -3 q^{-122} -6 q^{-126} -2 q^{-130} -2 q^{-132} + q^{-134} -4 q^{-136} -2 q^{-138} +8 q^{-140} -10 q^{-142} +6 q^{-144} -7 q^{-148} +14 q^{-150} -10 q^{-152} +6 q^{-154} +4 q^{-156} -5 q^{-158} +8 q^{-160} -3 q^{-162} - q^{-164} +6 q^{-166} -4 q^{-168} +3 q^{-172} -5 q^{-174} +6 q^{-176} -4 q^{-178} - q^{-180} + q^{-182} -3 q^{-184} +2 q^{-186} - q^{-188} + q^{-190} }[/math]

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 154"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ t^3-4 t+7-4 t^{-1} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+6 z^4+5 z^2+1 }[/math]

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 13, 4 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ q^{12}-2 q^{11}+2 q^{10}-3 q^9+2 q^8-2 q^7+2 q^6+q^3 }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ z^6 a^{-6} +6 z^4 a^{-6} +9 z^2 a^{-6} -2 z^2 a^{-8} -2 z^2 a^{-10} +4 a^{-6} -2 a^{-8} -2 a^{-10} + a^{-12} }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ z^8 a^{-10} +z^8 a^{-12} +z^7 a^{-9} +3 z^7 a^{-11} +2 z^7 a^{-13} +z^6 a^{-6} -5 z^6 a^{-10} -3 z^6 a^{-12} +z^6 a^{-14} -6 z^5 a^{-9} -15 z^5 a^{-11} -9 z^5 a^{-13} -6 z^4 a^{-6} -2 z^4 a^{-8} +7 z^4 a^{-10} -z^4 a^{-12} -4 z^4 a^{-14} -2 z^3 a^{-7} +9 z^3 a^{-9} +21 z^3 a^{-11} +10 z^3 a^{-13} +9 z^2 a^{-6} +5 z^2 a^{-8} -5 z^2 a^{-10} +2 z^2 a^{-12} +3 z^2 a^{-14} +3 z a^{-7} -3 z a^{-9} -10 z a^{-11} -4 z a^{-13} -4 a^{-6} -2 a^{-8} +2 a^{-10} + a^{-12} }[/math]